Exploiting a fundamental characteristic of galaxy assembly in the Λ cold dark matter paradigm, the observed spatial biasing and kinematics of metal-poor globular star clusters are used to constrain the local reionization epoch around individual galaxies. Selecting three galaxies located in different environments, the first attempt at constraining the environmental propagation of reionization in the local Universe is carried out. The joint constraint from the three galaxies (z reion = 10.5 +1.0/-0.9) agrees remarkably well with the latest Wilkinson Microwave Anisotropy Probe constraint on zreion for a simple instantaneous reionization model. More importantly, the range of zreion values found here is consistent with the global range of zreion estimates from other observations. We furthermore find a 1.7σ indication that reionization completed in low-density environments before the intergalactic medium in high-density environments was reionized. This is consistent with certain theoretical models that predict that reionization was globally prolonged in duration, with neutral hydrogen pockets surviving in high-density environments, even after the surrounding regions were reionized. More generally, this work provides a useful constraint on the formation history of galaxy stellar haloes.